Dynamical systems appear in many models in all sciences and in technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms.

Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results.

In these notes a sample of questions, for which the author is not aware of the existence of a good solution, are presented. Of course, it is easy to largely extend the list.

À. Haro and R. de la Llave,
A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.
doi: 10.1137/050637327.

À. Haro and R. de la Llave,
A parameterization method for the computation of invariant tori and their whiskers in quasiperiodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.
doi: 10.1137/050637327.